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The Mahonian probability distribution on words is asymptotically normal
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
2011 (English)In: Advances in Applied Mathematics, ISSN 0196-8858, E-ISSN 1090-2074, Vol. 46, no 1-4, 109-124 p.Article in journal (Refereed) Published
Abstract [en]

The Mahonian statistic is the number of inversions in a permutation of a multiset with a(i) elements of type i, 1 <= i <= m. The counting function for this statistic is the q analog of the multinomial coefficient (a(1) +...+4 a(m) a(1)...a(m)), and the probability generating function is the normalization of the latter. We give two proofs that the distribution is asymptotically normal. The first is computer-assisted, based on the method of moments. The Maple package Mahoni anStat, available from the webpage of this article, can be used by the reader to perform experiments and calculations. Our second proof uses characteristic functions. We then take up the study of a local limit theorem to accompany our central limit theorem. Here our result is less general, and we must be content with a conjecture about further work. Our local limit theorem permits us to conclude that the coefficients of the q-multinomial are log-concave, provided one stays near the center (where the largest coefficients reside).

Place, publisher, year, edition, pages
2011. Vol. 46, no 1-4, 109-124 p.
Keyword [en]
Mahonian statistics, Gaussian polynomials, Central and local limit theorem, Symbolic computation
National Category
URN: urn:nbn:se:uu:diva-154125DOI: 10.1016/j.aam.2009.10.001ISI: 000290190600009OAI: oai:DiVA.org:uu-154125DiVA: diva2:419302
Correction in Advances in Applied Mathematics 49(1):77doi: 10.1016/j.aam.2012.04.002Available from: 2011-05-26 Created: 2011-05-26 Last updated: 2012-07-19Bibliographically approved

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